With the mother gone, Duke proceeded with his plan to embezzle money from the company.
Literal
Mother [subject-が] gone [because-ので], Duke [topic-は] company [from-から] money [object-を] embezzle plan [object-を] advanced
いなくなった = 'became not present' (いる + なくなる). ヂュークis a katakana rendering of 'Duke' using the archaic ヂ (di) character. 横領する = to embezzle (横 side + 領 take = to take sideways/illegitimately). 計画をすすめた = 'advanced the plan.'